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 A090357 Satisfies A^5 = BINOMIAL(A)^4; also equals A090356^4. 3
 1, 4, 26, 244, 3131, 52600, 1111940, 28559320, 865622825, 30250881420, 1196941704454, 52860066623036, 2576115583371739, 137274420821505776, 7937914900025008984, 494941882189888642832, 33096552232229291234923 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS See comments in A090356. LINKS FORMULA G.f.: A(x)^5 = A(x/(1-x))^4/(1-x)^4. From Peter Bala, May 26 2015: (Start) O.g.f.: A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ), where b(n) = Sum_{k = 1..n} k!*Stirling2(n,k)*4^k = A094417(n) = 4*A050353(n) for n >= 1. BINOMIAL(A(x)) = exp( Sum_{n >= 1} c(n)*x^n/n ) where c(n) = (-1)^n*Sum_{k = 1..n} k!*Stirling2(n,k)*(-5)^k = A201365(n) = 5*A050353(n) for n >= 1. A(x) = B(x)^4 and BINOMIAL(A(x)) = B(x)^5 where B(x) = 1 + x + 5*x^2 + 45*x^3 + 495*x^4 + ... is the o.g.f. for A090356. See also A019538. (End) MATHEMATICA nmax = 16; sol = {a[0] -> 1}; Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^5 - A[x/(1 - x)]^4/(1 - x)^4 + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}]; sol /. Rule -> Set; a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *) PROG (PARI) {a(n)=local(A); if(n<1, 0, A=1+x+x*O(x^n); for(k=1, n, B=subst(A, x, x/(1-x))/(1-x)+x*O(x^n); A=A-A^5+B^4); polcoeff(A, n, x))} CROSSREFS Cf. A090356; A019538, A050353, A201365. Sequence in context: A210917 A210918 A052880 * A322766 A160886 A192546 Adjacent sequences: A090354 A090355 A090356 * A090358 A090359 A090360 KEYWORD nonn,easy AUTHOR Paul D. Hanna, Nov 26 2003 STATUS approved

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Last modified March 28 08:48 EDT 2023. Contains 361584 sequences. (Running on oeis4.)